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O R I G I NA L A RT I C L E
doi:10.1111/evo.12364
THE EVOLUTIONARY DYNAMICS OF SEXUALLY
ANTAGONISTIC MUTATIONS IN
PSEUDOAUTOSOMAL REGIONS OF SEX
CHROMOSOMES
Brian Charlesworth,1 Crispin Y. Jordan,1 and Deborah Charlesworth1,2
1
Institute of Evolutionary Biology, Ashworth Laboratory, School of Biological Sciences, The University of Edinburgh, King’s
Buildings, West Mains Road, Edinburgh EH9 3JT, United Kingdom
2
E-mail: [email protected]
Received October 10, 2013
Accepted January 8, 2014
Sex chromosomes can evolve gene contents that differ from the rest of the genome, as well as larger sex differences in gene
expression compared with autosomes. This probably occurs because fully sex-linked beneficial mutations substitute at different
rates from autosomal ones, especially when fitness effects are sexually antagonistic (SA). The evolutionary properties of genes
located in the recombining pseudoautosomal region (PAR) of a sex chromosome have not previously been modeled in detail. Such
PAR genes differ from classical sex-linked genes by having two alleles at a locus in both sexes; in contrast to autosomal genes,
however, variants can become associated with gender. The evolutionary fates of PAR genes may therefore differ from those of
either autosomal or fully sex-linked genes. Here, we model their evolutionary dynamics by deriving expressions for the selective
advantages of PAR gene mutations under different conditions. We show that, unless selection is very strong, the probability of
invasion of a population by an SA mutation is usually similar to that of an autosomal mutation, unless there is close linkage
to the sex-determining region. Most PAR genes should thus evolve similarly to autosomal rather than sex-linked genes, unless
recombination is very rare in the PAR.
KEY WORDS:
Gene expression, pseudoautosomal region, recombination, sex chromosomes, sexual antagonism.
The evolutionary fates of mutant alleles may differ between sex
chromosomes and autosomes. For example, in an XY system, a
male-benefit mutant allele of a Y-linked gene can spread even if
it is potentially lethal to females, whereas invasion of a population by a beneficial autosomal mutation requires a larger fitness
benefit to males than any cost to females. Most modeling work on
this aspect of sex chromosome evolution has focused on a fully
sex-linked region with no recombination in the heterogametic sex,
with a “genetically degenerate” Y chromosome that has lost nearly
all genes present on the X, so that males are generally hemizygous
for X-linked loci. In addition to any differences in effective population size that can affect the substitution rates of weakly selected
mutations, two major factors can cause differences between the
evolution of fully sex-linked regions and autosomes: opposing se-
1339
lection pressures on the two sexes (sexual antagonism, or “SA”),
and the dominance or recessivity of the fitness effects of adaptive
mutations; see reviews by Vicoso and Charlesworth (2006, 2009),
Connallon and Clark (2010), and Meisel and Connallon (2013).
For brevity, we will summarize previous results with reference to
XY systems; ZW systems mostly behave similarly, interchanging
males and females (in the Discussion, we mention situations in
which XY and ZW systems are expected to differ).
Under male hemizygosity, a new mutation in an X-linked
gene is present two-thirds of the time in females, but only onethird of the time in males. Thus, other things being equal, a fully
dominant female-benefit mutant allele at an X-linked locus can
invade a population unless male fitness is reduced by twice its
effect on females (Rice 1984). Favorable mutations with full or
C 2014 The Authors. Evolution published by Wiley Periodicals, Inc. on behalf of The Society for the Study of Evolution.
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Evolution 68-5: 1339–1350
B R I A N C H A R L E S W O RT H E T A L .
The genetic model. The figure shows an XY sex chromosome pair with a gene (indicated by a short vertical gray line) in
the PAR, at a genetic map distance of r from the male-determining,
Figure 1.
or male-specific region.
partial dominance at fully X-linked genes (with no Y-linked alleles) are therefore more likely than autosomal genes to evolve
enhanced female, relative to male, functions. However, strongly
recessive favorable mutations affecting males can be established
more easily on the X chromosome than the autosomes, because
of their greater exposure to selection in hemizygous males when
rare, and their fixation could result in the opposite pattern to that
just described (Rice 1984; Charlesworth et al. 1987). There has
been much discussion of how well these predictions are supported
by studies of genome-wide patterns of gene expression, which are
expected to correlate with effects on fitness (see the reviews cited
above; also Mank et al. 2010; Meisel et al. 2012).
The models just outlined have not considered loci with functional homologs on both the X and Y chromosomes. This situation can arise in fully sex-linked regions when there has not
been enough time for major genetic degeneration of the Y or
W chromosome. X-linked genes with functional Y-linked alleles
can also exist when the genetic degeneration of a Y-linked gene
is prevented by selection in the haploid stage of the life cycle,
for example, in predominantly haploid plants such as liverworts
(Okada et al. 2001), and in the male gametophytes in pollen in
flowering plants (Bergero and Charlesworth 2011; Chibalina and
Filatov 2011).
The recombining, partially sex-linked pseudoautosomal regions (PARs; see Fig. 1) of many extant XY or ZW systems
also carry functional copies on both sex chromosomes. Although
recombination continues, PAR genes will not undergo genetic
degeneration, and both sexes will have two alleles at PAR loci.
It is therefore of interest to understand how such genes will be
expected to evolve. The evolution of these regions has been little
studied (Otto et al. 2011), although it is clear that nonrecombining
regions of sex chromosomes have repeatedly evolved from states
with larger recombining PARs in Eutherian mammals (Lahn and
Page 1999; Skaletsky et al. 2003), birds (Lawson-Handley et al.
2004; Nishida-Umehara et al. 2007; Nanda et al. 2008; Pigozzi
2011), snakes (Matsubara et al. 2006; Vicoso et al. 2013a), and
plants (Bergero et al. 2007; Wang et al. 2012). New partially sex-
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linked genome regions can also arise by chromosome fusions or
translocations that add autosomal regions to the X chromosome,
provided that the added region continues to recombine with the
homologous autosome in males (reviewed in Bachtrog 2013).
The strength of selection for male- and female-benefit alleles
of partially sex-linked genes has not yet been studied quantitatively. We therefore examine the fates of new mutations in a PAR
of a sex chromosome system. We ask whether PAR genes differ
from autosomal loci in their tendency to fix alleles with sexspecific fitness effects (as fully sex-linked loci do). Specifically,
we relate the net strength of selection on SA mutations to the
frequency of recombination with the sex-determining region. The
model also yields results for fully X-linked and nondegenerated
Y-linked alleles, which is the extreme of the situation for PAR
genes, with no recombination between X and Y.
Our model is described in detail in the next section. It relates to two evolutionary situations that might result in sex differences in gene expression. A mutation at an SA locus may
affect gene expression differently in the two sexes (e.g., due to
their hormonal state or other consequences of their different genetic backgrounds), and may spread to intermediate frequencies
or fixation (Connallon and Clark 2010), leading directly to sexspecific gene expression. Alternatively, sex-specific gene expression may be a “secondary effect”, evolved in response to the
spread of alleles with SA effects (whether these arise from expression differences or in another way). Reduced expression of
such genes in males might then “resolve” the conflict and create
appropriate sex-specific gene expression (Rice 1984; Vicoso and
Charlesworth 2006); the resulting removal of the harmful fitness
effects of such mutations will result in their fixation, leaving a sex
difference in gene expression controlled by the second change.
The converse pattern of sex-specific change in expression would
occur with partially recessive mutations, as discussed above. In
either case, our question is the following: what portion of a PAR
should show a greater tendency to accumulate SA alleles, compared with autosomal loci? As one might expect intuitively, we
show that this effect is likely to be confined to PAR genes closely
linked to the boundary with the sex-determining region.
The Models and Analytical Results
We use exact and approximate analytic expressions to study the
invasion of populations by either male- or female-benefit alleles.
As in previous models of X-linked genes with male hemizygosity,
we assume that, for the fully sex-linked region, males are always
XY and females XX (i.e., the YY genotype is inviable). We study
selection coefficients of 0.1 or less, to which the approximations
used below should apply reasonably well. Both theory on the
fixation of mutations during “adaptive walks” (Orr 2005) and
E VO L U T I O NA RY DY NA M I C S O F S E X UA L LY A N TAG O N I S T I C M U TAT I O N S
evidence from studies of molecular evolution in Drosophila (Sella
et al. 2009; Schneider et al. 2011) suggest that small fitness effects
will predominate for favorable mutations.
We concentrate on the conditions for invasion of a population
by a newly arisen mutant allele, given the different forces acting
on it. We quantify the net selective advantage, σ, of a rare allele by
the asymptotic value of its rate of spread into the population (σ ∼
x/x, where x << 1 is the allele frequency in question). This
is closely related to the probability of survival of a new mutation
(which is equivalent to its probability of fixation when a balanced
polymorphism cannot be maintained). In the case of an autosomal
or fully sex-linked mutation, the survival probability is approximately 2σ in a Wright–Fisher population whose population size
N is sufficiently large that Ns > 1 (Haldane 1927; Charlesworth
et al. 1987). The asymptotic rate of spread ignores the need to
consider the three possible origins of mutations (X-linked in a female, X-linked in a male, and Y-linked in a male), whose survival
probabilities differ and should be appropriately averaged. However, the analysis described in the electronic supplementary material (ESM) shows that 2σ provides a good approximation to the
largest of the three survival probabilities (see Tables S1 and S2). It
also shows that, overall, our results described below overestimate
the effect of reducing recombination in increasing the net survival
probability of a new PAR mutation, even when allowing the possibility of a higher mutation rate in males than females (see ESM).
By deriving expressions for σ for both male- and female-benefit
alleles, we also obtain analytical approximations for the parameter
values under which balanced polymorphisms can be maintained.
MALE-BENEFIT MUTATIONS
The fitness scheme
We assume an invading male-benefit allele, A2 , and a resident
female-benefit allele, A1 . The recombination fraction between
the A locus and the sex-determining region is denoted by r
(Fig. 1). Using a modification of the fitness scheme of Jordan
and Charlesworth (2012), the fitnesses in the two sexes are specified by three parameters: s, the selection coefficient against the
less fit homozygous genotype in males (A1 A1 ); a constant, c,
that represents the relative strength of selection in females versus
males; the dominance coefficient, h, with respect to the fitness
effect of A1 in males and of A2 in females, such that 0 < h 1 (h here thus has a different meaning from that in Jordan and
Charlesworth 2012). To study invasion by A2 , only the ratios for
each sex of the fitness of A1 A2 to that of the resident homozygote
A1 A1 need be considered (see Table 1 and ESM). It is important
to note the difference between the net selective advantage σ and
s, the latter being the strength of selection acting on alleles at the
A locus.
The value of σ is obtained by subtracting one from the leading
eigenvalue of the matrix that describes the recursion relations for
The fitness model for an invading male-beneficial A2
allele in an XX/XY system.
Table 1.
Genotype
Males
Females
A1 A2
1 – hs
1 – chs
A1 A1
1–s
1
Under this scheme, 0 < c < 1 means that selection is stronger in males
than females, c = 1 means that it acts equally strongly in both sexes, and
c > 1 implies a greater “intensity” of selection in females than males. For
selection occurring only in males, c = 0; c = corresponds to selection only
in females (a ZW system can be modeled by exchanging males for females).
If c < 0, selection acts in the same direction in both sexes, which is of no
interest for the problem under considered here.
the frequencies of the various possible rare genotypes carrying
A2 , which are given in the ESM. Although the full characteristic
equation of this matrix is a cubic, and hence is hard to handle (Bull
1983, pp. 265–269), it can be reduced to a quadratic equation by
assuming that σ3 can be ignored (see the Appendix). If terms
higher than second order in s are also ignored in the relevant
solution, consistent with our assumption of weak selection, the
following simple expression for σ is obtained, provided that r is
not << s:
σ=
s2
s
[1 − h(1 + c)] + [3 − 2h(1 + c − 2hc) − h 2 (1 + c)2 ]
2
8
s2
+
(1)
[1 − h(1 − c)]2 + o(s 2 ).
16r
For r = ½, this is the same as the expression as for an autosomal
locus, where the terms in s2 reduce to ½s2 (1 – h).
This expression shows that σ is only weakly dependent on
r in this region of parameter space. The first two terms on the
right-hand side are independent of r, whereas the third term decreases with r. As expected intuitively, σ therefore increases as r
decreases. However, r only affects a term of order s2 ; recombination is thus likely have an important effect only when selection is
strong, except when 1 – h(1 + c) is of order s or less, in which
case σ is itself of order s2 in the range of r values in which this
approximation is valid.
A heuristic argument that also leads to this result is as follows.
When r and s are of comparable magnitude, the effect of linkage
on the selective advantage of an SA allele, increasing its value over
that with free recombination, is given by the product of its degree
of association with the sex that it benefits with its advantage in that
sex. The latter is O(s); the former is O(s/r), because the amount
of linkage disequilibrium in a two-locus system is of the order of
the ratio of the measure of epistasis in fitness (here of order s)
to the recombination rate (e.g., Charlesworth and Charlesworth
2010, p. 420).
When |1 – h(1 + c)| >> s, the second term on the right of
equation (1) can be ignored, and the equation can be rewritten in
EVOLUTION MAY 2014
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B R I A N C H A R L E S W O RT H E T A L .
terms of the scaled variables σ̃ = σ/s and r̃ = r/s, yielding the
following expression:
σ̃ =
1
1
[1 − h(1 + c)] +
[1 − h(1 − c)]2 + o(s).
2
16r̃
(2)
When this approximation is valid, σ̃ is a hyperbolic function
of r̃ , and its value depends on parameters that are independent of
the strength of selection, s. Because both 1 – h(1 + c) and the
magnitudes of the multiplier of 1/r̃ in equation (2) are of order 1,
σ̃ is unlikely to greatly exceed its minimum value unless r̃ is < 1,
that is, r is smaller than s.
When h(1 + c) – 1 >> s, the right-hand side of equation (2)
is negative, and so a loosely linked allele will not invade; invasion
by a male-beneficial allele is then possible only if the condition
for invasion at r = 0 is satisfied (given by eq. 4 below), and when r̃
is below a critical value. Setting σ̃ to 0 in equation (2), the critical
value is given by the following:
r̃cm ≈
[h(1 − c) − 1]
.
8[h(1 + c) − 1]
2
(3)
Because the magnitudes of the denominator and numerator
of this equation are both of order 1, the critical recombination
fraction is of order s, implying a requirement for very tight linkage
if s is small. As discussed in the ESM, the condition for σ to
exceed zero is equivalent to that for the survival probability of
the mutation to be nonzero, so that this result also applies to the
survival probability.
The above approach breaks down when r is much smaller than
s, but the behavior of s as a function of r can then be investigated
by evaluating ∂σ/∂r at r = 0 and using this derivative in the
Taylor series expansion of σ as a function of r, around its value
at r = 0 (see Appendix, eq. A6). The latter value can be found
as follows. With r = 0, male carriers of the invading allele have
fitness 1 – hs, whereas the fitness of the resident population is 1 – s.
If A2 is associated with the Y chromosome, the ratio of these two
fitnesses determines its rate of spread. To terms of order s, the
corresponding difference in fitness is s(1 – h), which is equivalent
to σ . When r is nonzero but small, we have the following:
σ̃ ≈ (1 − h) − r̃ .
(4)
If r = 0, this obviously requires the mutation to have arisen
on the Y chromosome. (The advantage to an X-linked mutation
is smaller than this, because it is present two-thirds of the time in
females, whose fitness is reduced by the mutation.) As r increases
away from zero, σ decreases linearly from this value if h < 1.
Equation (4) is valid even when the condition |1 – h(1 + c)|
>> s for the validity of equation (2) is not met. In this case, σ is
of order s2 at r = ½, and increases to approximately s(1 – h) at
r = 0, that is, σ increases (1/s)-fold. Thus σ should decrease with
increasing r over a wide range of r values, rather than becoming
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The fitness model for an invading female-beneficial A1
allele in an XX/XY system.
Table 2.
Genotype
Males
Females
A1 A2
1 – hs
1 – c´hs
A2 A2
1
1 – c´s
As before, c´ measures the relative strength of selection on males and females. In general, we may have c´ c, because only the ratios for each sex
of the fitness of the heterozygote to the fitness of the resident homozygote enter into the recursion relations (see ESM). For simplicity, however,
the equations presented in the text assume c´ = c, which is equivalent to
assuming similar levels of dominance in each sex. They can be made completely general by inserting c´ instead of c.
almost unchanging at a low r value, as occurs when |1 – h(1 + c)|
>> s; this is confirmed by numerical examples (see below).
FEMALE-BENEFIT MUTATIONS
In this case, an A1 allele invading an initially A2 A2 population
increases females’ fitness and may be deleterious in males. The
fitnesses of the heterozygotes and resident homozygotes of each
sex are displayed in Table 2; we will consider only the case
when the parameter c´ is equal to c. Once again, h measures the
dominance of the fitness effects of an allele in the sex whose
fitness is reduced by its presence. The fitness of A1 A2 females
relative to the resident A2 A2 population is now (1 – chs)/(1 – cs),
and that of A1 A2 males is 1 – hs. If σ for this case and for the
male-benefit case with the same set of parameters are both > 0, a
polymorphism will be maintained.
When r is not << s, a similar approach to that for a malebenefit mutation (see Appendix) results in the following approximate expression:
σ=
s2
s
[c(1 − h) − h] + [3c2 − 2hc(1 + c) − h 2 (1 + c)2 ]
2
8
s2
+
(5)
[c(1 − h) + h]2 + o(s 2 ).
16r
As in the case of a male-benefit mutation, σ decreases with
r, with the relevant term being of order s2 . The net selection
strength when r is ½ is again the same as that for an autosomal
locus. The condition for invasion with loose linkage that allows σ
to be positive and of order s is now c – h (1 + c) >> s.
When |c – h(1+ c)| >> s, we can use the same rescaling as
before to obtain
σ̃ ≈
1
1
[c(1 − h) − h] +
[c(1 − h) + h]2 .
2
16r̃
(6)
This yields the following critical value of r/s for invasion by
a female-benefit allele:
r̃c f ≈
[c(1 − h) − h]2
.
8[h(1 + c) − c]
(7)
E VO L U T I O NA RY DY NA M I C S O F S E X UA L LY A N TAG O N I S T I C M U TAT I O N S
(As for the male-benefit case, the condition for invasion with r =
0 must also be satisfied; this is given by eq. (8) below.)
When h > c/(1 + c) and h > 1/(1 + c), a balanced polymorphism will be maintained only when the recombination fraction
is smaller than the smaller of the two values given by equations
(3) and (7). For the relatively high recombination rates required
for equations (2), (3), (6), and (7) to be valid, low values of h (recessivity of the deleterious effects of the A locus mutant alleles)
favor invasion by both types of mutation; in contrast, a high value
of c (implying a stronger effect of the mutation in females than
males) favors invasion by female-benefit mutations, but has the
opposite effect on male-benefit mutations.
The value of σ when r is close to zero can be found
by the same method as before, giving the following first-order
approximation:
σ̃ =
[2c(1 − h) − h] r̃
− + o(s).
3
3
(8)
The first term of this expression gives σ̃ for the case of no
recombination. This is equivalent to the value obtained by weighting the sex-specific effects of selection for a completely X-linked
mutation by 2/3 for females and 1/3 for males, respectively, as is
appropriate for weak selection on an X-linked allele in the case
of a fully degenerate Y chromosome (Vicoso and Charlesworth
2006). The net strength of selection again declines linearly with
increasing recombination. Similarly to the male-benefit case, if
the condition |c – h(1 + c)| >> s is violated, there can be a
(1/s)-fold increase in σ between r = ½ and r = 0.
Comparing the conditions for invasion by male- and femalebenefit mutations with r near zero brings out an important contrast
with the autosomal case, as well as with X-linkage with a degenerate Y chromosome. Equation (4) implies that a weakly selected
male-benefit mutation will almost always be able to invade if the
recombination fraction is << s, unless h is very close to 1. This
means that it is very unlikely that a female-benefit mutation with
a deleterious effect on males will spread to fixation if it is closely
linked to the sex-determining region. Equation (8) implies that
a male-benefit mutation will be unable to spread to fixation if
h < 2c(1 – h); if c < ½, this is easily satisfied if h ½. This
is the main reason why polymorphisms for SA alleles are more
likely to be maintained in the PAR than on autosomes (Jordan and
Charlesworth 2012), particularly when they occur in loci closely
linked to its boundary with the nonrecombining region of the sex
chromosomes.
Numerical Results
We have generated numerical results for a range of cases, to test
the adequacy of the approximations made above, and to illustrate
the behavior of the system graphically. Figure 2 shows some
results for the invasion of a population by male-benefit alleles.
The parameters are defined in Table 1. We assumed that most
mutations have small fitness effects, and therefore chose three
values of s (which quantifies the selection against the initial male
genotype), with the highest value being s = 0.1; the figures show
only the s = 0.1 and 0.01 cases, because with s = 0.001 the
approximations agree very closely with the values calculated from
the full equation. The figures also show results for different values
of the parameter c, corresponding to stronger selection in males
than females (c < 1) and vice versa (c > 1), respectively. For the
harmful effects of the mutations in females, we assumed h = 0.5,
because intermediate dominance is plausible for mutations with
small fitness effects, and for mutations causing small changes
in expression levels (see Discussion, and the discussion of the
dominance of SA mutations in Connallon and Clark 2010). With
larger h, invasion is less likely than in the cases shown, but the
behavior is otherwise as described above.
The σ values calculated using the quadratic equation (A4)
agree well with those using the exact characteristic equation for
the system (eqs. A1 and A2), even when selection is strong (s =
0.1; the results are not shown, because the curves for the two cases
nearly coincide). With c = 0.5, Figure 2 shows values only for
recombination rates below the assumed values of s (except for the
r = ½ point), because, as can be seen from the figure, σ depends
very weakly on r for larger r values. The linear approximation
(4) for low r yields slightly lower values than the exact ones, but
again agreement is close when r is small. As expected, equation
(2) works well only for relatively loose linkage. With c = 2,
the male-benefit allele cannot invade the population unless r is
below a critical threshold value, and the examples shown in the
figure therefore focus on r values below this value, which is
well predicted by equation (A4) for both s values, but equation
(3) yields a value somewhat below the correct one. Equation (2)
does not work very well for the c = 2 case, because nonzero s
values require small r relative to s, which is where this expression
becomes inaccurate.
These results confirm that equations (2) to (4) can be used
to predict the fate of male-benefit mutations. When equation (3)
implies that there is no threshold r value, invasion is generally only
slightly more likely to occur for a partially sex-linked mutation
than for an autosomal one (in which r = ½). Figure 2 includes the
r = ½ case for the two selection coefficients modeled above for
c = 0.5. Only mutations closely linked to the male-determining
region, with r < s, show a strongly increased σ value. The σ
value is less than double that for an autosomal mutation unless
r < 0.0038 when s = 0.01; in the s = 0.1 case, this occurs when r <
0.036. For a 20% increase in σ, the corresponding r values for the
two cases are about 0.015 and 0.068, respectively. We therefore
conclude that PAR loci closely linked to the male-determining
region are expected to exhibit a higher rate of accumulation of
EVOLUTION MAY 2014
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B R I A N C H A R L E S W O RT H E T A L .
Figure 2.
Results for the invasion of a population by male-benefit alleles. The y-axis shows the net strength of selection (σ, see text),
and the x-axis is the frequency of recombination with the fully sex-linked region; and the r = ½ case represents autosomal loci. The
parameters are defined in Table 1. The figure shows examples of the s = 0.1 and 0.01 cases, with h = 0.5. Results are shown for two
values of the parameter c (0.5 and 2), where c < 1 corresponds to stronger selection in males than females and c > 1 to stronger selection
in females. In the former situation, the mutant allele can invade with any r value; in the latter situation, invasion occurs only if r is
below a threshold value (see text). Small dots are the values from equation (A4), large squares are from equation (2), and crosses from
equation (4).
SA mutations, but that most genes in the PAR are not. The same
conclusion applies to situations when there is a threshold r value;
in such cases, the outcome for r values above the threshold is the
same as for the autosomal case.
As noted in the previous section, when the condition |1 – h
(1 + c)| >> s for the validity of equation (2) is not met, equation
(1) must be used for the male-benefit case; the equivalent condition for the female-benefit case is when |c – h(1+ c)| >> s does
not hold, in which case equation (5) must be used. In such cases,
the expected dependence of σ on r is stronger at higher r values
than in the cases shown above. For comparison with the c = 2
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EVOLUTION MAY 2014
case in (Fig. 2), a numerical example for the male-benefit case is
shown in Figure 3, with s = 0.01, h = 0.5, and c = 1 (so that the
quantity above, as well the second term on the right of equation
(1), is zero). σ is again much larger for small r than for r = ½, but it
reaches double the autosomal value when r = 0.16, a much higher
recombination rate than seen in the cases shown above (but still
representing a small part of an entire PAR). From equation (1), it
can be seen that the r value required to generate a given ratio of σ
to its value at r = ½ does not depend on s when |c – h(1+ c)| << s,
so that an approximate doubling of σ should occur at r = 0.16 for
these values of h and c, over a wide range of s values.
E VO L U T I O NA RY DY NA M I C S O F S E X UA L LY A N TAG O N I S T I C M U TAT I O N S
Figure 3. Results for the invasion of a population by male-benefit
alleles in a situation when s = 0.01, h = 0.5, and c = 1, so that |1
– h(1 + c)| >> s, the condition for the validity of equation (2), is
not met. Small dots are the values from equation (A4), crosses are
from equation (1), and squares from equation (2).
This behavior arises because the derivatives of σ with respect
to r in equations (1) and (5) are of order s2 ; when s is small, they
therefore change very slowly until r is so small that these approximations break down, and the regions of validity of equation (4) or
(8) are approached, and σ becomes of order s instead of s2 . This
behavior, however, occurs only in a restrictive range of parameter
values, so that these cases can be regarded as exceptional.
The quadratic approximation for the female-benefit case
(substituting the expressions from equs. (A8) into eq. (A4), instead of eqs. (A2) for the male-benefit case) is also extremely
accurate, even for s = 0.1 (Fig. 4). shows cases with c = 0.75 and
c = 2, because, with the assumed h and s values, a mutation with
c = 0.5 does not invade. Again, the linear approximation (eq. (8)
for the female-benefit case) gives good agreement with the exact
value for r very close to zero. Equation (7) tends to overestimate
the critical r value, the opposite of what is seen in Figure 2 for
the male-benefit case. Again, therefore, the approximations can
be used to gain insights into the biological situation of interest,
and increased invasion probabilities (compared with autosomal
mutations) are expected only for mutations closely linked to the
male-determining region, unless the first term in equation (6) is
of order s2 .
tance from the sex-determining region (Jordan and Charlesworth
2012). This is relevant to the question of whether recombination
between X and Y (or Z and W) chromosomes has been suppressed
because of SA polymorphisms in the PAR (Charlesworth et al.
2005). Our study examines the conditions for the establishment
of SA mutations in PAR genes, concentrating on the spread of
weakly selected mutations, although we also provide analytical
results that predict when polymorphism will occur, which agree
well with deterministic numerical calculations of the full system
of genotype frequency equations for invasion by SA alleles. Although invasion conditions are not always sufficient to predict
when polymorphism will result, they do so for most biologically
plausible situations (Jordan and Charlesworth 2012).
To understand the extent to which PARs should display
genetic differences from autosomal regions, the conditions for
spread and/or fixation of SA mutations are of most interest, and
such differences are potentially detectable in empirical studies
(see below). Our study shows that the evolutionary dynamics of
genes across most of the PAR will generally differ little from that
of autosomal genes, unless it has a very short map length. Altered survival probabilities of SA mutations in PAR genes should
not extend beyond genes closely linked to the boundary with the
fully sex-linked region. Although we model XY systems, corresponding results apply to ZW systems, reversing the sexes; some
distinctive properties of ZW systems are discussed below.
In addition, the qualitative difference between recessive and
dominant mutations due to hemizygosity in the heterogametic sex
has created difficulties for empirical tests of the predictions for
fully sex-linked genes (see the introductory section). For genes
with functional copies on both X and Y chromosomes, however,
both male- and female-benefit SA mutations are disfavored if
they have dominant deleterious fitness effects. Invasion of maleversus female-benefit SA mutations should therefore not depend
as strongly on their level of dominance as in the case of fully
sex-linked, male-hemizygous genes, except for mutations with
fully recessive fitness effects. However, our results showing that
most PAR genes should not evolve differently from autosomal
loci suggests that empirical studies of PAR genes will not help to
solve the problem of the effects of dominance on the outcome of
SA selection.
PREDICTED CHANGES IN EXPRESSION DURING SEX
CHROMOSOME EVOLUTION
Discussion
THE EVOLUTIONARY DYNAMICS OF MUTATIONS IN
FULLY AND PARTIALLY SEX-LINKED GENES
The main question previously studied in relation to PAR gene
evolution was whether SA polymorphisms can be maintained, and
the extent to which this is facilitated by a low recombination dis-
Most theory for sex-specific expression patterns on X chromosomes and autosomes (see Introduction) assumes that the fixation of male-benefit mutations will lead to higher levels of
gene expression in males, relative to females (and vice-versa
for female-benefit SA mutations). This “resolution” of SA effects by modifiers of expression will also promote the fixation of
polymorphisms unless they are fully Y-linked. Our study of initial
EVOLUTION MAY 2014
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B R I A N C H A R L E S W O RT H E T A L .
Figure 4.
Results for the invasion of a population by female-benefit alleles. The figure is organised and labeled as in Figure 2, but the
two values of the c parameter are 0.75 and 2, because with c = 0.5 the mutant never invades. Small dots are the values from equation
(A4), crosses are from equation (8), and squares from equation (6).
invasion conditions of SA mutations thus helps to predict whether
PAR genes should evolve sex differences in gene expression, regardless of whether fixation or polymorphism occurs. They imply
that the evolution of sex-specific patterns of gene expression in the
PAR is not likely to differ greatly from that for autosomal genes,
except for loci that are very closely linked to the sex-determining
region or subject to strong selection.
DIFFERENCES BETWEEN ZW AND XY SYSTEMS
In many species, sexual selection is stronger in males than females, potentially generating larger fitness differences among
males than females, and thus leading to male-benefit mutations
becoming established more easily than female-benefit mutations,
other things being equal. In addition, invasion conditions are more
1346
EVOLUTION MAY 2014
stringent for female- than male-benefit SA mutations in the XY
case, and vice versa for ZW (unless selection is much stronger
on males than females), as pointed out previously (Jordan and
Charlesworth 2012). This is because male-benefit mutations become strongly associated with the Y-linked male-determining
region when linkage is tight, allowing them to be commoner in
males than females (with r = 0, eq. (4) is identical in form to
that for a Y- [or W-] linked mutation with no corresponding X
or Z homolog), whereas, even if a female-benefit allele is highly
associated with the X chromosome, it will often be present in
males. Male-benefit PAR mutations with small r are thus more
likely to become fixed in XY systems, and female-benefit genes
in ZW systems, implying that the evolution of male- rather than
female-biased PAR expression for such genes is more likely to
E VO L U T I O NA RY DY NA M I C S O F S E X UA L LY A N TAG O N I S T I C M U TAT I O N S
occur in XY systems, and vice versa for ZW systems, although
strong sexual selection on males could overcome this effect for
ZW systems.
EMPIRICAL RESULTS FOR PAR GENES
It is very difficult to find out whether the sex chromosomes present
in a given lineage, including their PARs, have fixed unexpectedly
many male- or female-benefit mutations compared with autosomal genes, and to determine whether they have SA effects. One
approach has been to study sex differences in patterns of gene
expression. It seems reasonable to assume that such expression
differences, excluding the effects of dosage compensation, imply
past sex differences in effects on fitness, or extant unresolved
ones (Vicoso and Charlesworth 2006; Mank et al. 2008, 2010;
Meisel et al. 2012), because the fixation of a male-benefit allele
that harms females favors reduced expression in females, leading
to the evolution of higher levels of expression in males relative
to females (Rice 1984; Vicoso and Charlesworth 2006), and vice
versa for female-benefit SA alleles. This is thought to contribute
to the so-called masculinization of X chromosomes, along with
the accumulation of genes with male reproductive functions (e.g.,
Vicoso and Charlesworth 2006; Meisel et al. 2012; Mueller et al.
2013).
Complete Y linkage restricts male-benefit alleles to males,
partially resolving conflicts between the effects of PAR gene polymorphisms in males versus females, although the expression of
X-linked female-benefit alleles can still harm males. Alternatively, rapid evolution of sex-specific expression could resolve
conflicts, removing selection to reduce recombination in a PAR.
It has recently been suggested that the evolution of sex-specific
expression could explain the persistence of ancient PARs, as is
found in sexually monomorphic ratite birds such as emus (Vicoso
et al. 2013b).
However, it is often suggested that weak sexual selection reduces the chance of selective differences between the two sexes,
and hence of SA effects. Non-sexually dimorphic birds, such as
the polyandrous emu (Coddington and Cockburn 1995), should
thus be less likely than other birds to evolve sex-specific expression of PAR genes. Comparative studies might be worthwhile, to
test for associations between sexual selection or dimorphism, and
sex-specific expression of PAR genes, but will require information from many species about their sex chromosomes and levels
of sexual selection. In the three-spine stickleback, Gasterosteus
aculeatus, which is highly sexually dimorphic, no evidence was
found for sex differences in expression of 80 PAR genes studied
(Leder et al. 2010). We are not aware of any other comparable data, and PAR gene expression should be studied in other
species. However, important aspects of sexual selection may not
remain the same over the timescale of evolution of gene expression. Changes in mating systems have occurred in the ratites, such
as rheas (Bruning 1973) and ostriches (Bertram 1992), which are
polygynous.
Assuming, however, that SA effects are important in emus,
can their gene expression patterns be explained by the evolutionary dynamics of PAR mutations? An estimated 20 to 50% of
genes in the emu PAR show higher male expression bias than
autosomal genes (Vicoso et al. 2013b). This suggests the fixation of male-benefit alleles with small effects on fitness at many
PAR genes, followed by the evolution of lower expression in females in response to SA effects (or simply fixations of alleles
with higher expression in males, associated with enhanced male
fitness). Switching the sexes, so that female-benefit alleles in the
XY case correspond to male-benefit alleles in the emu ZW case,
our results imply that at least 20% of the genes studied must be
closely linked to the fully W-linked female-specific region. Unfortunately, there is currently no genetic map for the emu, nor any
information about the locations of genes in the PAR in relation to
the sites of crossing over in meiosis. However, if the physically
large emu PAR indeed occupies a small genetic map length, this
would suggest that this species has evolved partial recombination
suppression.
Overall, it thus remains unclear why some lineages evolve
sex-specific expression of SA alleles in partially sex-linked genes,
whereas others reduce recombination with the fully sex-linked
region. Possibly, some lineages simply have a low genetic variance
for recombination rates, and cannot respond to selection favoring
suppressed recombination.
CONCLUSIONS AND FUTURE POSSIBILITIES
A major reason for studying the models described here is to develop predictions that may generate tests for the importance of
sexually antagonistic fitness effects of mutations. Ideally, direct
evidence is needed for trade-offs between the effects of mutations
on different organismal functions, and detailed genetic information, such as the proportion of loci involved and the estimated
strengths of selection. Although there is currently considerable
evidence for SA effects in species with long-established separate sexes and sex chromosomes (e.g., Connallon and Knowles
2005; Mank and Ellegren 2009), this is indirect. No conclusive
examples of such genes are known, other than some cases of sexual selection in fish species (Fisher 1930; Lindholm and Breden
2002; Gordon et al. 2012). Evidence for the differential accumulation of genes with sex-biased expression might be a further
source of evidence, using genome regions in which this is predicted. Alternatively, tests for balanced polymorphisms due to
SA selection could use recently evolved sex chromosomes with
extensive PARs. In the flowering plant Silene latifolia, a study
of sequence polymorphisms suggested high diversity in the PAR
relative to other genomic regions, consistent with such a situation
EVOLUTION MAY 2014
1347
B R I A N C H A R L E S W O RT H E T A L .
(Qiu et al. 2013). Such studies have not yet been related to patterns
of evolution of DNA sequences, or of gene expression.
ACKNOWLEDGMENTS
We thank Mark Kirkpatrick and an anonymous reviewer for their comments on the previous version of this article. CYJ was supported by
BBSRC grant Bb/J006580/1 to R.A. Ennos.
DATA ARCHIVING
The doi for our data is 10.5061/dryad.t1m2q.
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Associate Editor: S. Glemin
Appendix
ANALYSIS OF THE CHARACTERISTIC EQUATIONS FOR
INVASION BY MALE- OR FEMALE-BENEFIT
MUTATIONS
Male-benefit case
Following Bull (1983, pp. 265–269), if second-order terms in
the frequencies of genotypes carrying the invading allele A2 are
ignored, we obtained a three-dimensional matrix describing the
recursion relations. The characteristic equation of this matrix has
the form
λ3 + pλ2 + q = 0,
(A1)
where
p=−
q=
(1 − r )(1 − hs) (1 − hsc)
−
,
(1 − s)
2
(A2a)
(1 − hsc)(1 − hs)2 (1 − 2r )
.
2(1 − s)2
(A2b)
Writing σ = λ – 1, and including terms in σ 2 but neglecting those
in σ 3 , equation (A1) can be reduced to a quadratic equation:
(3 + p)σ2 + (3 + 2 p)σ + 1 + p + q = 0.
(A3)
This can be approximated to o(s2 ) by using the second derivatives
of p and q given in the ESM. Using the argument described there,
the first term on the right of equation (A5) can be written as
s
s2
s2
[1 − h(1 + c)] + (1 − h)(1 − hc) + [1 − h(2 + c)
2
2
4r
+ h 2 (1 + c + c2 )] + o(s 2 ).
The second term can be approximated by
−
s2
(3 + p)[1 − h(1 + c)]2 + o(s 2 )
8r
s2
= −
(3 + 2r )[1 − h(1 + c)]2 + o(s 2 ).
16r
The final second-order approximation for σ is obtained by adding
these two terms, yielding equation (1) of the main text.
The behavior of σ as a function of r when r is small can be
investigated by using implicit differentiation of equation (A3) to
determine ∂σ/∂r . This gives
(1 + 2σ + σ2 ) ∂∂rp + ∂q
∂σ
∂r
=−
.
∂r
2(3 + p)σ + 3 + 2 p
(A6)
Using equations (A2), and the expressions for σ, p, and q when
r = 0, the numerator of the fraction on the right-hand side for r =
0 can be written as
[2s(1 − h)(1 − s) + s 2 (1 − h)2 + (1 − s)2 − (1 − s)(1 − hs)(1 − hsc)](1 − hs)
.
(1 − s)3
For small s, the leading term in this expression is s[1 + h (c – 1)].
The leading term in the denominator is equal to 3s(1 – h). This
yields equation (4) of the main text.
A similar procedure can be used to determine ∂σ/∂r when
r = ½, which provides a check on the approximations used above.
In this case, the selection coefficient for an invading A2 allele is ½
s[1 – h(1+c)] + ½ s2 (1 – h) + o(s2 ). Substituting this into equation
(A6), together with the appropriate expressions for p and q, we
obtain the final result:
∂σ
s2
= − [1 − h(1 − c)]2 + o(s 2 ).
(A7)
∂r r=1/2
4
This has the following relevant solution, which is accurate to
terms of order s2 :
−(3 + 2 p) + (3 + 2 p)2 − 4(3 + p)(1 + p + q)
. (A4)
σ=
2(3 + p)
This agrees with the expression found by differentiating equation
(1) at r = ½.
This can be approximated further as follows, noting that the term
under the square root sign can be written as (3 + 2p)2 [1 – 4(3
+ p)(1 + p + q)/(3 + 2p)2 ]; this is close to (3 + 2p)2 . Taking
the first two terms in the Taylor series for the bracketed term, we
obtain the second-order approximation
This case gives the following terms in the characteristic equation
(A3):
σ≈−
(1 + p + q) (3 + p)(1 + p + q)2
.
−
(3 + 2 p)
(3 + 2 p)3
(A5)
Female-benefit case
p = −(1 − r )(1 − hs) −
q=
(1 − hsc)
,
2(1 − sc)
(1 − hsc)(1 − hs)2 (1 − 2r )
.
2(1 − sc)
EVOLUTION MAY 2014
1349
(A8a)
(A8b)
B R I A N C H A R L E S W O RT H E T A L .
The derivatives of p and q used for the approximations for the
selective advantage σ of A2 are given in the ESM. Using a similar
argument to that for the male-beneficial case, the approximations
shown in the main text can be derived.
Supporting Information
Additional Supporting Information may be found in the online version of this article at the publisher’s website:
Table S1. Male-benefit mutations with male heterogamety.
Table S2. Female-benefit mutations with male heterogamety.
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EVOLUTION MAY 2014